In an interview with The Washington Post's Mike Musgrove, Nintendo of America president Reggie Fils-Aime handily dismissed the DS's competition, suggesting either that he doesn't feel threatened at all, or he feels really threatened (and the history of DS sales makes us more likely to believe the former interpretation).
He said that the newly launched, download-only PSP Go has a "fundamental concept problem in terms of 'Who's it for?' and 'What's the benefit?'" He added that he has the "utmost respect for all our competitors, but it's interesting to try and answer the consumer question of 'What's in it for me?' in that product."
As for the other major competition in the handheld space? When discussing the DS's advantages over the iPhone/iPod Touch platform, Fils-Aime made the surprising move of referring to a third-party game, namely Scribblenauts. "That's a fabulous experience that can only be brought to life on the DS." Honestly, it probably could be done on iPhone, but it hasn't and probably won't, which is just as good for Nintendo.